相关论文: Perfectoid covers of abelian varieties

We prove that Shimura varieties of abelian type with infinite level at $p$ are perfectoid. As a corollary, the moduli spaces of polarized K3 surfaces with infinite level at $p$ are also perfectoid.

For an abelian variety $A$ over an algebraically closed non-archimedean field $K$ of residue characteristic $p$, we show that the isomorphism class of the pro-\'etale perfectoid cover $\widetilde A=\varprojlim_{[p]}A$ is locally constant as…

We give a simple characterization of all perfectoid profinite \'{e}tale covers of abelian varieties in terms of the Hodge-Tate filtration on the $p$-adic Tate module. We also compute the geometric Sen morphism for all profinite $p$-adic Lie…

The tilting correspondence is a fundamental property of perfectoid fields. In this note, we show that the tilting construction can also be used to detect perfectoid fields among nonarchimedean fields. In particular, for $K$ a complete…

In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of…

There has been a long-standing question about whether being perfectoid for an algebra is local in the analytic topology. We provide affirmative answers for the algebras (e.g., over $\overline{\mathbb{Z}_p}$) whose spectra are inverse limits…

We show that infinitesimal Torelli for $n$-forms holds for abelian covers of algebraic varieties of dimension $n\ge 2$, under some explicit ampleness assumptions on the building data of the cover. Moreover, we prove a variational Torelli…

Perfect quadratic forms give a toroidal compactification of the moduli space of principally polarized abelian g-folds that is Q-factorial and whose ample classes are characterized, over any base. In characteristic zero it has canonical…

Perfectoid spaces have become a crucial tool in $p$-adic geometry, serving as a bridge between adic spaces in characteristic $0$ and those in characteristic $p$. In this article, we develop a systematic way to study the structure of…

We show that, for an abelian variety defined over a $p$-adic field $K$ which has potential good reduction, its torsion subgroup with values in the composite field of $K$ and a certain Lubin-Tate extension over a $p$-adic field is finite.

Let $k$ be an algebraic extension of $\mathbb F_p$ and $K/k$ a regular extension of fields (e.g. $\mathbb F_p(T)/\mathbb F_p$). Let $A$ be a $K$-abelian variety such that all the isogeny factors are neither isotrivial nor of $p$-rank zero.…

We show that up to potential isogeny, there are only finitely many abelian varieties of dimension $d$ defined over a number field $K$, such that for any finite place $v$ outside a fixed finite set $S$ of places of $K$ containing the…

We show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields,…

We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $\mathbb Z[\pi,q/\pi]$-ideal and a…

Deligne's weight-monodromy conjecture gives control over the poles of local factors of L-functions of varieties at places of bad reduction. His proof in characteristic p was a step in his proof of the generalized Weil conjectures. Scholze…

In this paper we determine the number of isomorphism classes of superspecial abelian varieties $A$ over the prime field $\Fp$ such that the relative Frobenius morphism $\pi_A$ satisfying $\pi_A^2=-p$.

We affirmatively answer a conjecture in the preprint ``Essential dimension and algebraic stacks,'' proving that the essential dimension of an abelian variety over a number field is infinite.

An abelian variety over a field K is said to have big monodromy, if the image of the Galois representation on l-torsion points, for almost all primes l contains the full symplectic group. We prove that all abelian varieties over a finitely…

The endomorphism ring End(A) of an abelian variety A is an order in a semi-simple algebra over Q. The co-index of End(A) is the index to a maximal order containing it. We show that for abelian varieties of fixed dimension over any…

The main purpose of this paper is to prove the following theorem on the defect relations for ample divisors of abelian varieties. Main Theorem. Let $A$ be an abelian variety of complex dimension $n$ and $D$ be an ample divisor in $A$. Let…